Criteria for accepting an invitation to become an editor of a scientific journalI think the following criterion is somewhat rational: when the publishers of a new journal solicit help for the editorial board, they understand that people will be asking themselves exactly the kind of question you are asking. Therefore they make an effort to establish the reputation of the new journal in the email - providing references to editors who have already joined, explaining why a new journal in this field is called for, contrasting it with similar journals etc. On the other hand they don't say anything about the submission process since this is completely irrelevant.

Examples of eventual counterexamplesPretty impressive! Specifically, gcd(n^17+9, (n+1)^17+9)=1 for all n up to some crazy explicit number, the number of digits of which I couldn't even count. This begs the question, is there a reasonably simple proof that this gcd isn't always 1?

Feb17

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Describe a tree by junctionsPerhaps it's just me, but I'm completely unable to parse this. Could you perhaps explain what you mean by sectors, infinite branches, junctions and what is the tree? A picture, perhaps?

Unique factorization in polynomial ringsRead literally, one cannot even state (let alone prove) #1 without having a notion of a "field" which I imagine would disqualify both Euclid and Guass. The first general definition of a field is by Weber (1891) according to Wikipedia. Earlier notions were things like a subfield of the complex numbers. I'm not sure if the question assumes that the prover knew they were working over general fields, or rather looking for proofs which are essentially independent of the base field (even if they were formulated over a specific field).